Chapter 13 National Science Foundation Systemic Initiatives
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Mathematics in African History and Cultures
Paulus Gerdes & Ahmed Djebbar MATHEMATICS IN AFRICAN HISTORY AND CULTURES: AN ANNOTATED BIBLIOGRAPHY African Mathematical Union Commission on the History of Mathematics in Africa (AMUCHMA) Mathematics in African History and Cultures Second edition, 2007 First edition: African Mathematical Union, Cape Town, South Africa, 2004 ISBN: 978-1-4303-1537-7 Published by Lulu. Copyright © 2007 by Paulus Gerdes & Ahmed Djebbar Authors Paulus Gerdes Research Centre for Mathematics, Culture and Education, C.P. 915, Maputo, Mozambique E-mail: [email protected] Ahmed Djebbar Département de mathématiques, Bt. M 2, Université de Lille 1, 59655 Villeneuve D’Asq Cedex, France E-mail: [email protected], [email protected] Cover design inspired by a pattern on a mat woven in the 19th century by a Yombe woman from the Lower Congo area (Cf. GER-04b, p. 96). 2 Table of contents page Preface by the President of the African 7 Mathematical Union (Prof. Jan Persens) Introduction 9 Introduction to the new edition 14 Bibliography A 15 B 43 C 65 D 77 E 105 F 115 G 121 H 162 I 173 J 179 K 182 L 194 M 207 N 223 O 228 P 234 R 241 S 252 T 274 U 281 V 283 3 Mathematics in African History and Cultures page W 290 Y 296 Z 298 Appendices 1 On mathematicians of African descent / 307 Diaspora 2 Publications by Africans on the History of 313 Mathematics outside Africa (including reviews of these publications) 3 On Time-reckoning and Astronomy in 317 African History and Cultures 4 String figures in Africa 338 5 Examples of other Mathematical Books and 343 -
An Alternative Route to Teaching Fraction Division: Abstraction of Common Denominator Algorithm
International Electronic Journal of Elementary Education, 2015, 7(3), 399-422. An Alternative Route to Teaching Fraction Division: Abstraction of Common Denominator Algorithm İsmail Özgür ZEMBAT Mevlana (Rumi) University, Turkey Received: February 2015 / Revised: May 2015 / Accepted: May 2015 Abstract From a curricular stand point, the traditional invert and multiply algorithm for division of fractions provides few affordances for linking to a rich understanding of fractions. On the other hand, an alternative algorithm, called common denominator algorithm, has many such affordances. The current study serves as an argument for shifting curriculum for fraction division from use of invert and multiply algorithm as a basis to the use of common denominator algorithm as a basis. This was accomplished with the analysis of learning of two prospective elementary teachers being an illustration of how to realize those conceptual affordances. In doing so, the article proposes an instructional sequence and details it by referring to both the (mathematical and pedagogical) advantages and the disadvantages. As a result, this algorithm has a conceptual basis depending on basic operations of partitioning, unitizing, and counting, which make it accessible to learners. Also, when participants are encouraged to construct this algorithm based on their work with diagrams, common denominator algorithm formalizes the work that they do with diagrams. Keywords: Teaching fraction division, abstracting common denominator algorithm, curriculum development Introduction Arithmetic operations, and teaching and learning of them have always been an interest for mathematics education community. In his historical analysis, Usiskin (2007) pointed out that operations (especially on fractions) still preserve its importance in school mathematics and they should be given enough emphasis. -
AVAILABLE from ABSTRACT DOCUMENT RESUME Exemplary
DOCUMENT RESUME ED 434 033 SE 062 877 TITLE Exemplary and Promising Mathematics Programs. INSTITUTION Department of Education, Washington, DC. PUB DATE 1999-00-00 NOTE 71p.; Prepared by the "Math and Science Education Expert Panel." AVAILABLE FROM ED Pubs, P.O. Box 1389, Jessup, MD. Tel: 877-433-7827 (Toll Free); Web site: <http://www.enc.org>. PUB TYPE Reports Descriptive (141) EDRS PRICE MF01/PC03 Plus Postage. DESCRIPTORS *Academic Standards; Demonstration Programs; Elementary Secondary Education; Mathematics Curriculum; *Mathematics Education; *Program Descriptions; *Program Evaluation ABSTRACT Selecting programs, textbooks, and curriculum materials is one of the most important decisions educators make. An Expert Panel on Mathematics and Science was established by the U.S. Department of Education to develop a high-quality, research-based process for selecting programs and to use that selecting process to identify exemplary and promising programs. This booklet describes eight mathematics programs designated as exemplary or promising. A detailed explanation of the submission and selection process and an explanation of how the programs were separated into exemplary and promising categories is included. The four characteristics used for the selection criteria include quality of program, usefulness to others, educational significance, and evidence of effectiveness and success. Each characteristic has multiple criteria with specific indicators of achievement. Each program is described in terms of the four selection characteristics and professional -
May 2001 What Are the Children Learning and Who Decides
May 2001 THE TEXTBOOK CONUNDRUM What are the Children Learning and Who Decides? Introduction With the quality of education in the United States now the biggest domestic concern, demands for accountability are in vogue. The drive for better schools however, is usually limited to issues of standards, testing, choice and teachers. Missing from most discussions is the role that textbooks play in the achievement of children. A few facts to consider: • In more than twenty states, the state (state board of education, department of education, secretary or commissioner of education, or another specially designated state textbook committee) picks the textbooks for every classroom in the state – either through outright text selection, or recommendations from a short list. To control curriculum, they tie funding to compliance with the states’ textbook adoption policy. • Textbooks supplied to three states, California, Texas and Florida – all of which give significant influence to state agencies for textbook selection – account for 30 percent of the more than $3.3 billion K-12 textbook market in 1998, the most recent year for which statistics are available.i • Four publishers (McGraw-Hill, Houghton Mifflin, Harcourt, and Pearson) control 70 percent of the industry. Size means money means influence in the textbook world. They are a strong, quiet interest group that works behind the scenes and through major education groups to ensure that the process favoring them stays exactly the way it is. The process for putting books in front of children then, looks something like this: The “big three” states draw up textbook adoption policies to which the “big four” publishers try to align their textbook content. -
The Impact of Information and Communication Technology on Our Understanding of the Nature of Mathematics
FLM 26(1) 2/2/06 12:33 PM Page 29 THE IMPACT OF INFORMATION AND COMMUNICATION TECHNOLOGY ON OUR UNDERSTANDING OF THE NATURE OF MATHEMATICS FREDERICK K. S. LEUNG The incorporation of information and communication tech- scientific calculator) and it can also be used as a transforma- nology (ICT) into mathematics education constitutes one of tional tool, a data collection and analysis tool, a visualizing the most important themes in contemporary mathematics tool and/or a checking tool. Doerr and Zangor (2000) sum- education. For example, one of the six principles of school marized the different roles the graphing calculator plays and mathematics as espoused by the National Council of Teach- the corresponding student actions as follows: ers of Mathematics (NCTM, 2000) is that Role of the Description of Student Actions Technology is essential in teaching and learning math- Graphing Calculator ematics; it influences the mathematics that is taught and enhances students’ learning. (p. 24, emphasis added). Computational Tool Evaluating numerical expressions, esti- mating and rounding The purpose of this article is to review research findings on the effectiveness of the use of some selected ICT tools in Transformational Tool Changing the nature of the task mathematics teaching and learning. [1] This review is then Data Collection and Gathering data, controlling phenomena, set against a summary of the actual use of ICT tools in real Analysis Tool fiding patterns classrooms. Various issues related to the incorporation of Visualizing Tool Finding symbolic functions, displaying ICT into mathematics education arise, including: data, interpreting data, solving equations • the reasons for the discrepancy between the poten- Checking Tool Confirming conjectures, understanding tial and actual use of ICT multiple symbolic forms Figure 1: Table showing the patterns and modes of use of a • the impact of ICT on our understanding of the graphing calculator (p. -
Traditional Mathematics Teaching Is Still the Norm in Our Nation's Schools
Forum on Public Policy Mathematics and Literature: Educators’ Perspectives on Utilizing a Reformative Approach to Bridge Two Cultures Suzanne J. Nesmith, Associate Dean/Assistant Professor of Education, Wayland Baptist University Abstract The existence of two distinct cultures within our society, the arts and the sciences, was introduced by physicist C. P. Snow in his 1959 Reed Lecture at Cambridge University and was further illuminated by Snow in The Two Cultures, a Second Look (1964). Lamenting the existence of the cultural chasm while also fearing a widening of the schism, Snow opened a dialogue with the hope of reconciliation between the cultures. Fifty years later, although the chasm still exists, numerous entities and individuals within the cultures work to formulate the means by which the cultures may be bridged, and, subsequently, the chasm narrowed and eliminated. The National Council of Teachers of Mathematics (NCTM) is one such entity, and leaders of the group have worked since the council‟s establishment in 1920 to research, promote, and improve the teaching and learning of mathematics through innovative, reformative approaches. Reform-oriented curricula are built upon constructivist perspectives aimed at assisting students in utilizing their own unique backgrounds and experiences to develop a personal understanding of mathematical situations. One means of infusing personal experience into the mathematics curriculum, while also bridging the aforementioned cultures, is through the incorporation of children‟s literature, yet there exists great variance in the type, format, structure, and success of the methodology‟s implementation. Subsequently, while the reformative approach of mathematics literature integration presents as a means of building understanding by bridging the cultures of mathematics and the humanities, it is the educator‟s choices relevant to the approach which have the greatest impact on the outcomes of the approach. -
Review of Evaluation Studies of Mathematics and Science Curricula and Professional Development Models
Abstract This report identifies mathematics and science curricula as well as professional development models at the middle and high school levels that are effective based on their success in increasing student achievement. The goal of the study was to provide some choice to districts and schools that wanted guidance in selecting a curriculum and that wished to use effectiveness as a selection criterion. Unexpectedly, most middle and high school mathematics and science curricula did not have studies of student achievement with comparison groups, and it proved especially difficult to find effects in either math or science for subgroups by sex, minority status, and urban status. Findings strongly suggest that science curricula is more effective when it is inquiry-based, although math curricula can be effective when standards- or traditional-based. REVIEW OF EVALUATION STUDIES OF MATHEMATICS AND SCIENCE CURRICULA AND PROFESSIONAL DEVELOPMENT MODELS By Beatriz C. Clewell Clemencia Cosentino de Cohen The Urban Institute and Patricia B. Campbell Lesley Perlman Campbell-Kibler Associates, Inc. with Nicole Deterding Sarah Manes Lisa Tsui The Urban Institute and Shay N.S. Rao Becky Branting Lesli Hoey Rosa Carson Campbell-Kibler Associates, Inc. Submitted to the GE Foundation December 2004 Acknowledgments A number of individuals contributed to this effort in various ways. We were fortunate to have the assistance of Gerhard Salinger of the National Science Foundation; Jo Ellen Roseman of Project 2061 at the American Association for the Advancement of Science (AAAS); Joan Abdallah at AAAS; and several staff members at the Center for Science Education at the Education Development Center—Barbara Berns, Jeanne Century, Joe Flynn, Elisabeth Hiles, Jackie Miller, Marian Pasquale, and Judith Sandler —in helping us identify science curricula that might have evaluation studies. -
The Comparison of Physical/Virtual Manipulative on Fifth-Grade Students’ Understanding of Adding Fractions
Running Head: COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE The Comparison of Physical/Virtual Manipulative on Fifth-Grade Students’ Understanding of Adding Fractions A dissertation submitted to the Graduate School of the University of Cincinnati in partial fulfillment of the requirements for the degree of Doctor of Education in the Department of Curriculum and Instruction of the College of Education by Sami Alshehri M.A. Umm Al-Qura University July 2008 Committee Chair: M. Sally, Ed.D. COMPARISON OF PHYSICAL/VIRTUAL MANIPULATIVE ABSTRACT The purpose of this quantitative study was to compare two types of manipulatives in order to see their effects upon understanding of adding fractions for three comparable groups of fifth grade students. A total of 163 students who demonstrated low mathematical performance participated in the project in order to learn the addition of fractions by using physical and virtual manipulatives for the experimental groups and the normal mathematic curriculum for the control group. The intervention occurred during a two-week time frame in six public elementary schools in Abha, Saudi Arabia where students used fraction bars for both physical and virtual manipulatives in order to build conceptual understanding of adding fractions properly. Instructions were provided to all the participants directly regarding what the participants were to do each day of the 2-week experiment. Pre-and post-tests, an attitude survey, and a preference survey were the instruments that were used to collect data during the study. A repeated measures design with a cross over treatment was used for comparing the effects of the two modes of treatments, virtual and physical manipulatives, compared to a control group for the understanding of adding fractions for the three groups of students. -
Reforming Mathematics Education Dr. Barry Fagin Professor Of
Reforming K-12 Mathematics Education Dr. Barry Fagin Professor of Computer Science USAFA (opinions are mine, not USAFA’s) What’s the problem? 8th grade math scores, IAEEA 2000 Report 700 600 500 400 300 200 100 0 y n a a a A ia n ca re a i ri S n ra ri o p ss ysi a U a rke I f p u la lg u A a Ja a m T g R u o th in M B R u S o S 1. Singapore 2. Korea 3. China 13. Australia 4. Hong Kong 2. Czechoslovakia 5. Japan 3. Finland 1. Moldova 6. Belgium 4. Malaysia 2. Thailand 7. Netherlands 5. Bulgaria 3. Israel 8. Slovak Rep. 6. USA 4. Tunisia 9. Hungary 7. England 5. Macedonia 10.Canada 8. New Zealand 6. Turkey 11.Slovenia 9. Lithuania 7. Jordan 12.Russia 10.Italy 8. Iran 11.Cyprus 9. Indonesia 12.Romania 10.Chile 11.Philippines 12.Morocco 13.South Africa % CSAP Advanced or Proficient (5th grade) 100 90 80 70 60 W 50 H 40 B 30 20 10 0 2001 2002 2003 % CSAP Advanced or Proficient (8th grade) 100 90 80 70 60 W 50 H 40 B 30 20 10 0 2001 2002 2003 % CSAP Advanced or Proficient (10th grade) 100 90 80 70 60 W 50 H 40 B 30 20 10 0 2001 2002 2003 Class of 2005 100 90 80 70 60 W 50 H 40 B 30 20 10 0 8th (2001) 9th (2002) 10th (2003) What’s the cause? Widespread adoption of curricula that: • Emphasize process over content • De-emphasize math facts • Embrace constructivism • Encourage calculator use in early grades • Encourage group work How did this happen? Open letter of protest published in Post, Carter elected DOE forms expert signed by > 200 panel to identify mathematicians and DOE created noteworthy math scientists programs Reagan elected, Secretary -
Math Wars: the Politics of Curriculum
University of Northern Iowa UNI ScholarWorks Presidential Scholars Theses (1990 – 2006) Honors Program 1999 Math wars: The politics of curriculum Raymond Johnson University of Northern Iowa Let us know how access to this document benefits ouy Copyright ©1999 Raymond Johnson Follow this and additional works at: https://scholarworks.uni.edu/pst Part of the Curriculum and Instruction Commons, and the Science and Mathematics Education Commons Recommended Citation Johnson, Raymond, "Math wars: The politics of curriculum" (1999). Presidential Scholars Theses (1990 – 2006). 89. https://scholarworks.uni.edu/pst/89 This Open Access Presidential Scholars Thesis is brought to you for free and open access by the Honors Program at UNI ScholarWorks. It has been accepted for inclusion in Presidential Scholars Theses (1990 – 2006) by an authorized administrator of UNI ScholarWorks. For more information, please contact [email protected]. MATH WARS The Politics of Curriculum Presidential Scholars Senior Thesis by Raymond Johnson Under the advisement of Dr. Edward Rathmell December 9, 1999 Johnson 2 Introduction There is an ongoing battle in mathematics education, a battle sometimes so fierce that some people call it the "math wars". Americans have seen many changes and proposed changes in their educational system in the last fifty years, but few have stirred such debate, publicity, and criticism as have the changes in mathematics education. We will first look at the history of the math wars and take time to examine previous attempts to change mathematics education in America. Second, we will assess the more recent efforts in mathematics education reform . Lastly, we will make some predictions for the future of the math wars and the new directions mathematics education may take. -
What Students Notice As Different Between Reform and Traditional Mathematics Programs
Journal for Research in Mathematics Education 2008, Vol. 39, No. 1, 9–32 What Students Notice as Different Between Reform and Traditional Mathematics Programs Jon R. Star Harvard University John P. Smith III Michigan State University Amanda Jansen University of Delaware Research on the impact of Standards-based mathematics and reform calculus curricula has largely focused on changes in achievement and attitudes, generally ignoring how students experience these new programs. This study was designed to address that deficit. As part of a larger effort to characterize students’ transitions into and out of reform programs, we analyzed how 93 high school and college students perceived Standards-based and reform calculus programs as different from traditional ones. Results show considerable diversity across and even within sites. Nearly all students reported differences, but high-impact differences, like Content, were not always related to curriculum type (reform or traditional). Students’ perceptions aligned moderately well with those of reform curriculum authors, e.g., concerning Typical Problems. These results show that students’ responses to reform programs can be quite diverse and only partially aligned with adults’ views. Key words: College/university; Curriculum; High school (9–12); Integrated curriculum; Learners (characteristics of); Qualitative methods; Reform in mathematics education; Social factors One inevitable outcome of the development of new curricula in any field, but perhaps particularly in mathematics, is to provoke discussion and debate about what content can and should be taught in schools and universities. The recent cycle of The entire Mathematical Transitions Project team played a vital role in the work reported here. In addition to the authors, the other members of this team were, in alphabetical order, Dawn Berk, Carol Burdell, Beth Herbel-Eisenmann, Gary Lewis, and Violeta Rosca-Yurita. -
Quantitative Literacy: Why Numeracy Matters for Schools and Colleges, Held at the National Academy of Sciences in Washington, D.C., on December 1–2, 2001
NATIONAL COUNCIL ON EDUCATION AND THE DISCIPLINES The goal of the National Council on Education and the Disciplines (NCED) is to advance a vision that will unify and guide efforts to strengthen K-16 education in the United States. In pursuing this aim, NCED especially focuses on the continuity and quality of learning in the later years of high school and the early years of college. From its home at The Woodrow Wilson National Fellowship Foundation, NCED draws on the energy and expertise of scholars and educators in the disciplines to address the school-college continuum. At the heart of its work is a national reexamination of the core literacies—quantitative, scientific, historical, and communicative — that are essential to the coherent, forward-looking education all students deserve. Copyright © 2003 by The National Council on Education and the Disciplines All rights reserved. International Standard Book Number: 0-9709547-1-9 Printed in the United States of America. Foreword ROBERT ORRILL “Quantitative literacy, in my view, means knowing how to reason and how to think, and it is all but absent from our curricula today.” Gina Kolata (1997) Increasingly, numbers do our thinking for us. They tell us which medication to take, what policy to support, and why one course of action is better than another. These days any proposal put forward without numbers is a nonstarter. Theodore Porter does not exaggerate when he writes: “By now numbers surround us. No important aspect of life is beyond their reach” (Porter, 1997). Numbers, of course, have long been important in the management of life, but they have never been so ubiquitous as they are now.